Читать книгу Quantum Physics is NOT Weird - Paul J. van Leeuwen - Страница 31

A vibrating string has only very specific ways in which a standing wave can arise and continue. The condition is that an integer number of half wavelengths fits exactly on the length of the string. Which means ½λ, 1λ, 1½λ, 2λ, 2½λ, etc. You get the lowest vibration – the tonic – if exactly ½λ fits on the string. With 1λ fitting on the string you get the 1st harmonic, with 1½λ fitting on the string you get the 2nd harmonic, and so on. This means that there will always be a lowest vibration for any given string, to evoke a lower tone is impossible. The possible vibrations are clearly ‘quantized’ and have a lowest boundary. This phenomenon inspired prince Louis Victor de Broglie to a rather daring idea.

Figure 4.9: Seven harmonically vibrating modes of a string. From tonic to 6^th harmonic.

Source: Wikimedia Commons.

Thinking about Bohr's still enigmatic atomic model and familiar with Einstein's photons – EM-waves behaving as particles – Louis Victor de Broglie (1892-1987) connected in 1923 the dots and supposed that if a wave could be a particle, the other way around might also be true. Could it be possible that those enigmatic Bohr electron orbits were standing electron waves?

If electrons behaved like standing waves, then their waves would only fit on their orbits around the nucleus in certain ways, just like standing waves on vibrating strings. When the electron wavelength fitted exactly on circular orbits of 1 λ, 2 λ, 3 λ, etc. this would result in standing electron waves.

De Broglie knew very well that a moving particle, such as an electron, has a certain momentum p = m.v, where m.v denotes mass times speed. Combining this with the Planck formula for the energy of a light particle and Einstein’s energy-mass equivalence, he proposed that this determined the associated wavelength of a moving electron.

It was already confirmed by the photoelectric effect ^[21] that a photon has an associated momentum. Combining E = h.f, E = m.c2 and p = m.v De Broglie arrived at a simple formula for the wavelength of an electron: λ = h/(mv). This tells that the greater the speed v is, the shorter the associated electron wavelength λ will be. When jumping to a higher orbit the speed of the electron will be slowed down and its wavelength will therefore grow accordingly.

Figure 4.10: The first four De Broglie harmonic electron orbits in the hydrogen atom.

Figure 4.10 shows the De Broglie standing electron waves, from left to right, from the lowest orbit, the "tonic" (n = 1), up to and including the 4th orbit (n = 4), the 3rd ‘harmonic’. Incidentally, I hope you will notice that these De Broglie orbits differ from standing waves on a vibrating string with fixed ends. One full wavelength fits on the lowest orbit, two full wavelengths fit on the next orbit, etc. Standing wave orbits with other than an integer number of wavelengths are not possible. This has to do with the fact that after having traveled a full orbit, the moving wave front must arrive exactly in phase with its ‘tail’ to interfere constructively. The electron wave just goes around and is not reflected on a fixed boundary as does happen on a vibrating string. Reflecting on fixed boundaries invokes 180^o phase shifts which correspond to half wavelengths. In short, only an integer number of wavelengths are allowed for circular standing waves ^[22].

Only four years later De Broglie's hypothesis became confirmed experimentally, albeit in a rather circuitous way. Coincidence – serendipity – played a major role.

The wave character of electrons was demonstrated in 1927 by Clinton Davisson (1881-1958) and Lester Germer (1896-1971) in experiments where electrons were reflected on a crystal lattice of nickel atoms. The results of their experiment, preferred directions in which the scattered electrons reflected, could only be explained with interference – which implied wave behavior. This is in fact the same line of thought that Young followed with his double-slit experiment in 1805. Interference presumes waves.

Two years after the confirmation of electron waves by Davisson and Germer, Louis De Broglie received the physics Nobel Prize in 1929. A well-earned recognition for his daring out-of-the-box thinking. For a good understanding of what the Davisson-Germer experiment entailed and what it confirmed, we should first pay some attention to diffraction grids.